Diversification and Non-Ergodicity – A layman’s perspective

The argument for diversification is ancient wisdom and pre-dates the financial markets as we know them today.  After all, the age old saying, “Don’t put all your eggs in one basket”, was first recorded in a 1605 Spanish publication.  It’s a simple, yet useful idea that has been drilled into generation after generation.

As academic disciplines age, however, there is a tendency to repackage simple and intuitive ideas and shroud them in a cloud of complex jargon and high mathematics (physics envy perhaps?).  In this process, without even realising, we sometimes end up justifying the right ideas with flawed reasoning. I realised this when I came across a presentation connecting the idea of diversification with something called non-ergodicity.

Like any layman, my first reaction was – what in the world is ERGODICITY???  In a nutshell, ergodic systems are those where “the average outcome of the group is the same as the average outcome of the individuals comprising that group over time”.  In a non-ergodic system, on the other hand, the results of the group diverge from the results of the individuals that constitute that group over time[1].

The bet

To set the context, the author uses the example of a “favourable” bet of successive coin tosses with the following rules (note the double quotes on the word favourable as we will come back to it later):

• You start with a capital of $100;
• If the coin lands heads, you win 50%; and
• If it lands tails, you lose 40%.

As the expected return on this bet is positive, it is a “favourable bet” and one where you should keep betting your entire capital on each successive coin toss.   According to the author, the results from this bet are counter intuitive such that if an individual keeps playing this bet, he would invariably go bankrupt.  This is despite the “favourable” nature of the bet.  To test this outcome, I ran some simulations and turns out, it is true for the most part.  Based on this, the author reaches the conclusion that reason this is happening is because the player is betting his entire capital on each coin toss, which will sooner or later make an individual go bankrupt despite the “favourable” nature bet.  Taking the lessons from this reasoning and applying to investing, the author suggests that an investor should not have highly concentrated bets since doing that is likely to result in eventual bankruptcy even if the bets are favourable.

While I agree that diversification is a key risk management tool, I disagree with the reasoning used to reach the above conclusion.  In my view, if what the author is saying is true then, this proposition should hold true no matter how favourable the bet really is.  For instance, an individual playing this game should go bankrupt betting his entire capital even if he loses 30% on a tails instead of 40%.  However, that is not the case.  Attached is a simple worksheet where you can change the parameters of the bet and see for yourself.

In my view, the reason an individual is going bankrupt in the author’s example is because the bet is not favourable in the first place.  IT IS AN UNFAVOURABLE BET.  Let me explain why.  The probability of a coin toss landing heads/ tails is 50% and therefore, to understand the true payoff from this bet, we should toss the coin twice to see the resulting capital at the end of two coin tosses.  Below are the results:

Scenario	Starting Capital ($)	Ending Capital ($)	Gain/ (Loss)
HH	100	225	+125%
HT/ TH	100	90	-10%
TT	100	36	-64%

 

Given that the probability of landing a heads or tail is 50%, the most likely scenario over two coin tosses is Scenario 2, i.e. a heads followed by a tails or a tails followed by a heads.  The ending capital in this scenario is $90, i.e. less than the beginning capital of $100.  Now you see?  In the most likely scenario, the bet is unfavourable where you end up with a lower capital than what you started with.  This is the reason why you would eventually go bankrupt playing this game.

So am I against diversification?

Now, I am not at all recommending that you should go out and bet your entire capital on what may seem like a favourable bet. After all, doing that would be equivalent to playing the lottery and even though you might win one, I think sensible people already know the likely outcome if you keep playing it long enough.

As I said in the beginning, I am a strong believer in reasonable diversification.  A technical explanation simply is that up to a point, diversification reduces your exposure to firm-specific and tail risks and it more than offsets the reduced return potential.  However, there are other reasons for reasonably diversifying your portfolio as well:

• In investing, the rules of the game and the range of possible outcomes are not as tightly defined as a simple coin toss.In fact, there are various unknowns and variables impacting each company which are impossible for the human mind to a) comprehend, and b) to correctly judge the consequent behaviour of economic participants.  The economy is an adaptive system and we still don’t know exactly how the it works; and
• You don’t get to play the same game over and over again. Each company that you invest in represents a different bet with different pay-offs and probabilities. Thus, you might get a coin toss with a certain set of rules one day and the next day you might end up playing a throw of a die with different sets of rules and different probabilities.

There is another, more fundamental reason why diversification is important – most of our expectations about the future are built on analysing past data.  And while the past can be a useful guide, there is always the possibility that it may not be relevant anymore.  If there is one lesson that year 2020 has taught us, it is this – S*@T HAPPENS.  Ultimately, you can’t win a game that you can’t play.  Intelligent diversification is intended to allow you to keep playing the game.

[1] This article contains a detailed explanation of the concept of ergodicity.